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It is True:
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h(x) is the same as f(x) and g(x), they are all concave functions.
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We can apply hypothesis testing on two sets of samples to test whether they are independent or related no matter what PDF do they obey. Before start a parametric hypothesis testing, we need to know the statistic mean of the sample set. But Central Limit Theorem holds especially true for sample sizes over 30. So the sample size should be noticed because there are not enough observations on both RV: each of them contains ten observations. As a result, non-parametric hypothesis testing should be considered.
Considering two sets of car prices as two sequences and applying Mann-Whitney Test of Independence is a good hypothesis testing opinion on both equal or unequal size of sample sets.
Assume that when deciding whether an applicant is low-risk (c0), medium-risk (c1), or high-risk (c2) for bank loans, the bank only considers the amount of money in the saving accounts of the applicants. Historic data indicate following distribution about the savings (in thousand dollars) of the applicant based on their classes:
(x|c0) ∼ Uniform(35,60), (x|c1) ∼ Uniform(15,45), (x|c2) ∼ Uniform(0,20)
Furthermore, we know that 20% and 30% of the previous applicants are low-risk and medium-risk, respectively. If an applicant has 48,000 dollars in her savings account, predict the applicant’s class (based on Bayes decision rules).
Hint: pdf for uniform distribution — if x ∼ Uni f orm(a,b), f(x) = 1/(b−a) f or a <= x <= b.
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$\color{red}{\text{P(c0|x)=0.336}}$
$\color{red}{\text{P(c1|x)=0.332}}$
$\color{red}{\text{P(c2|x)=0.332}}$
for:
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as we know:
$f′(x) = 3x^2 + 6$
$f′′(x) = 6x $
$f''(x)>0,\ when\ x>0\ \therefore$ f(x) is convex at x when $x>0$
$f''(x)<0,\ when\ x<0\ \therefore$ f(x) is concave at x when $x<0$
(Hint :To find the critical values put $f ′(x) = 0$)
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critical values are at $f ′(x) = 0$
$f ′(x) = 3x^2 + 6 = 0$
$ x^2 = -2$
$ x = \pm\sqrt{2}i$